Advertisements
Advertisements
प्रश्न
Write the value of \[\left[ \hat {i} - \hat {j} \hat {j} - \hat {k} \hat {k} - \hat {i} \right] .\]
Advertisements
उत्तर
We have
\[\left[\hat { i} - \hat {j} \hat {j} - \hat {k} \hat {k} - \hat {i} \right] = \left[ \left( \hat {i} - \hat {j} \right) \times \left( \hat {j} - \hat {k} \right) \right] \cdot \left( \hat{k} - \hat {i} \right) \left( \because \left[ \vec{a} \vec{b} \vec{c} \right] = \left( \vec{a} \times \vec{b} \right) . \vec{c} \right)\]
\[ = \left[ \left( \hat {i} \times \hat {j} \right) - \left( \hat {i} \times \hat {k} \right) - \left(\hat { j} \times \hat {j} \right) + \left( \hat {j} \times \hat {k} \right) \right] \cdot \left(\hat { k} - ]\hat {i} \right)\]
\[ = \left[ \hat {k} + \hat {j} + \hat {i} \right] \cdot \left(\hat { k} - \hat {i} \right)\]
\[ = \left[ \left( \hat {k} \cdot \hat {k} \right) - \left( \hat {k} \cdot\hat { i } \right) + \left( \hat {j} \cdot \hat {k} \right) - \left( \hat {j} \cdot \hat {i} \right) + \left( \hat {i} \cdot \hat {k} \right) - \left( \hat {i} \cdot \hat {i} \right) \right]\]
\[ = 1 - 0 + 0 - 0 + 0 - 1 = 0\]
APPEARS IN
संबंधित प्रश्न
Show that the four points A(4, 5, 1), B(0, –1, –1), C(3, 9, 4) and D(–4, 4, 4) are coplanar.
Find the value of λ, if four points with position vectors `3hati + 6hatj+9hatk`, `hati + 2hatj + 3hatk`,`2hati + 3hatj + hatk` and `4hati + 6hatj + lambdahatk` are coplanar.
Prove that a necessary and sufficient condition for three vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that \[l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\]
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 11 \hat{i} , \vec{b} = 2 \hat{j} , \vec{c} = 13 \hat{k}\]
Show of the following triad of vector is coplanar:
\[\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k} , \vec{b} = -\hat{ i} + 4 \hat{j} + 3 \hat{k} , \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = \hat {i} + 3 \hat {j} , \vec{b} = 5 \hat {k} , \vec{c} = \lambda \hat {i} - \hat {j}\]
Prove that: \[\left( \vec{a} - \vec{b} \right) \cdot \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\} = 0\]
\[\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{and} \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},\]
If c1 = 1 and c2 = 2, find c3 which makes \[\vec{a,} \vec{b} \text { and } \vec{c}\] coplanar.
If four points A, B, C and D with position vectors 4 \[\hat { i} +3\] \[\hat { j} +3\] \[\hat { k} ,5\] \[\hat { i} +\] \[x\hat { j} +7\] \[\hat { k} ,5\] \[\hat { i} +3\] \[\hat { j}\] and \[7 \hat{i} + 6 \hat{j} + \hat{k}\] respectively are coplanar, then find the value of x.
Write the value of \[\left[ \hat {i} + \hat {j} \ \hat {j} + \hat {k} \ \hat {k} + \hat {i} \right] .\]
Find the volume of the parallelopiped with its edges represented by the vectors \[\hat {i} + \hat {j} , \hat {i} + 2 \hat {j} \text { and } \hat {i} + \hat {j} + \pi k .\]
If \[\vec{a,} \vec{b}\] \[\text { are non-collinear vectors, then find the value of} \left[ \vec{a} \vec{b}\hat { i} \right] \hat{i} + \left[ \vec{a} \vec{b} \hat {j} \right] \hat {j} + \left[ \vec{a} \vec{b} \hat {k} \right] \hat {k} .\]
For any two vectors \[\vec{a} \text { and } \vec{b}\] of magnitudes 3 and 4 respectively, write the value of \[\left[ \vec{a} \vec{b} \vec{a} \times \vec{b} \right] + \left( \vec{a} \cdot \vec{b} \right)^2 .\]
Find \[\vec{a} . \left( \vec{b} \times \vec{c} \right)\], if \[\vec{a} = 2 \hat {i} + \hat {j} + 3 \hat {k} , \vec{b} = - \hat {i} + 2 \hat {j} + \hat {k}\] and \[\vec{c} = 3 \hat { i} + \hat {j} + 2 \hat {k}\].
If \[\vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = 0\] for some non-zero vector \[\vec{r} ,\] then the value of \[\left[ \vec{a} \vec{b} \vec{c} \right],\] is
If \[\vec{a,} \vec{b,} \vec{c}\] are non-coplanar vectors, then \[\frac{\vec{a} \cdot \left( \vec{b} \times \vec{c} \right)}{\left( \vec{c} \times \vec{a} \right) \cdot \vec{b}} + \frac{\vec{b} \cdot \left( \vec{a} \times \vec{c} \right)}{\vec{c} \cdot \left( \vec{a} \times \vec{b} \right)}\] is equal to
If the vectors \[4 \hat { i} + 11 \hat {j} + m \hat {k} , 7 \hat { i} + 2 \hat { j} + 6 \hat {k} \text { and } \hat {i} + 5 \hat {j} + 4 \hat {k}\] are coplanar, then m =
If \[\vec{a,} \vec{b,} \vec{c}\] are three non-coplanar vectors, then \[\left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} + \vec{c} \right) \right]\] equals
Find the angle between the lines whose direction cosines l, m, n satisfy the equations 5l + m + 3n = 0 and 5mn − 2nl + 6lm = 0.
Find `bar"a".(bar"b" xx bar"c")` if `bar"a" = 3hat"i" - hat"j" + 4hat"k" , bar"b" = 2hat"i" + 3hat"j" - hat"k"` and `bar"c" = - 5hat"i" + 2hat"j" + 3hat"k"`
If `bar"u" = hat"i" - 2hat"j" + hat"k" , bar"v" = 3hat"i" + hat"k"` and `bar"w" = hat"j" - hat"k"` are given vectors, then find `(bar"u" + bar"w").[(bar"u" xx bar"v") xx (bar"v" xx bar"w")]`
If `vec"a" = hat"i" - 2hat"j" + 3hat"k", vec"b" = 2hat"i" + hat"j" - 2hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"`, find `vec"a" * (vec"b" xx vec"c")`
The volume of the parallelepiped whose coterminus edges are `7hat"i" + lambdahat"j" - 3hat"k", hat"i" + 2hat"j" - hat"k", -3hat"i" + 7hat"j" + 5hat"k"` is 90 cubic units. Find the value of λ
If `vec"a", vec"b", vec"c"` are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of `(vec"a" + vec"b") * (vec"b" xx vec"c") + (vec"b" + vec"c")* (vec"c" xx vec"a") + (vec"c" + vec"a") * (vec"a" xx vec"b")`
If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k", hat"i" + hat"k"` and `"c"hat"i" + "c"hat"j" + "b"hat"k"` are coplanar, prove that c is the geometric mean of a and b
The volume of tetrahedron whose vertices are A(3, 7, 4), B(5, -2, 3), C(-4, 5, 6), D(1, 2, 3) is ______.
If the volume of the tetrahedron formed by the coterminous edges `bar"a", bar"b" and bar"c"` is 5, then the volume of the parallelopiped formed by the coterminous edges `bar"a" xx bar"b", bar"b" xx bar"c" and bar"c" xx bar"a"` is
If `veca = hati + hatj + hatk, veca.vecb` = 1 and `veca xx vecb = hatj - hatk`, then find `|vecb|`.
Determine whether `bb(bara and barb)` are orthogonal, parallel or neither.
`bar a = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`
If `bar"u" = hat"i" - 2hat"j" + hat"k" , bar"v" = 3hat"i" + hat"k"` and `bar"w" = hat"j" - hat"k"` are given vectors, then find `[bar"u" xx bar"v" bar"u" xx bar"w" bar"v" xx bar"w"]`
If a vector has direction angles 45ºand 60º find the third direction angle.
If `u=hati -2hatj + hatk, barr=3hati + hatk and w=hatj, hatk` are given vectors, then find `[baru + barw]. [(barw xx barr)xx(barr xx barw)]`
Determine whether `bara and barb` are orthogonal, parallel or neither.
`bara = -3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5 hati + 4 hatj + 3 hatk`
Find the volume of a tetrahedron whose vertices are A(−1, 2, 3) B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).
